Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.4.49
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 2 to ∞) 1 / eᵏ
Verified step by step guidance1
Identify the series given: \( \sum_{k=2}^{\infty} \frac{1}{e^k} \). This is a series where each term is \( \frac{1}{e^k} \).
Recognize that this is a geometric series because the terms can be written as \( \left( \frac{1}{e} \right)^k \), where the common ratio \( r = \frac{1}{e} \).
Recall the geometric series test: a geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if and only if \( |r| < 1 \).
Check the value of \( r = \frac{1}{e} \). Since \( e \approx 2.718 \), \( \frac{1}{e} < 1 \), so the series converges by the geometric series test.
To find the sum of the series starting from \( k=2 \), use the formula for the sum of a geometric series starting at \( k=0 \): \( S = \frac{a}{1-r} \), then adjust for the starting index by subtracting the first terms as needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio r. It converges if the absolute value of r is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term.
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Geometric Series
Convergence Tests for Series
Convergence tests help determine whether an infinite series converges or diverges. Common tests include the Ratio Test, Root Test, and Comparison Test, which analyze the behavior of terms or ratios to conclude about the series' convergence.
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07:51Choosing a Convergence Test
Exponential Functions in Series
Exponential functions like e^k grow rapidly, and their reciprocals (1/e^k) decrease exponentially. Recognizing such terms helps apply convergence tests effectively, as series with terms decreasing exponentially often converge.
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6:13Exponential Functions
Related Practice
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
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Textbook Question
35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)
Textbook Question
Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}
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